【Mathathon3】Asymptotic Mean Value Expansion for Solutions of General Ellipticand Palabolic Equations

Speaker: Prof Juan Manfredi

Title:Asymptotic Mean Value Expansion for Solutions of General Ellipticand Palabolic Equations

Abstract:

The classical mean value property characterizes harmonic functions. It can be extended to characterize solutions of many linear equations. We will focus in an asymptotic form of the mean value property that characterizes solutions of nonlinear equations. This question has been partially motivated by the connection between Random Tug-of-War games and the normalized \(p-\) Laplacian equation discovered some years ago, where a nonlinear asymptotic mean value property for solutions of a PDE is related to a dynamic programming principle for an appropriate stochastic game. Our goal is to show that an asymptotic nonlinear mean value formula holds for several types of non-linear elliptic equations.
Our approach is flexible and allows us to consider several families of operators obtained as an infimum, a supremum, or a combination of both infimum and supremum, of linear operators. We study both when the set of coefficients is bounded and unbounded (each case requires different techniques). Examples include Pucci, Issacs, Monge-Ampère, and \(k-\) Hessian operators and some of their parabolic versions.
This talk is based in joint work with Pablo Blanc (Buenos Aires), Fernando Charro (Detroit), and Julio Rossi (Buenos Aires).

Profile:

Manfredi works on elliptic and parabolic partial differential equations of p-Laplacian type, including the case p equals infinity, in Euclidean spaces and in subRiemannian manifolds. These equations are used to model phenomena where the relevant energy is non-quadratic. He is interested in regularity and other fine properties of p-harmonic functions.